74 
NATURE 
[May 26, 1870 
Spottiswoode, V.P., having taken the chair) gave an account of 
his paper ‘‘ On the Mechanical Description of a nodal bicircular 
Quartic.” Take a quadrilateral OAA‘O’, in which the adjacent 
Cc 
(e} 
0 0 
sides OA, AA’ are equal to each other, and the other two 
adjacent sides OO’, O'A’ are also equal to each other. O, O! 
are fixed points, and we have thus a link, AA’, the extremi- 
ties of which are connected with the radii OA, O'A’, respec- 
tively, and consequently describe circles about the centres O, O’ 
respectively, the radius OA of the one circle being equal to the 
length AA’ of the link, and the radius O/A' of the other 
circle being equal to the distance OO’ of the centres. The 
theorem is that any point C rigidly connected with the link AA’ 
describes a nodal bicircular Quartic; that is, a quartic curve 
with three nodes, two of the nodes being the circular points at { 
infinity. Any such curve is the inverse of a conic, and it is 
also the antipode of aconic. For the analytical investigation, 
the origin is taken at O, the axis of « in the direction OO’, 
and the axis of y at right angles thereto, upwards {rem O. The 
inclinations of OA, AA’, O'A’ to the axis Ox are taken to be 
0, >, 6 respectively. Also write OA = AA’= a, AB = 4, 
BC =c, OO'= O’A’ = a’, and make m = 7 Se Zthen 
ata 
we haye for the locus of C— : 
x =acos@+ dbcosp—csing 
y=asin@ + dsing + ccos >; 
since 6! 6+ ¢. Also we have tan® tan “ =m; 
bs =i 0 
whence writing tan — = “, we have tan 2 = — and the 
2 2 u 
equation to the locus may be written— 3 } 
: w—1 u? — mi? 2mu | 
iP ae 2 rast 3 
ue oT ue +m wtm | 
eee 5 2m 2° — mi? 
a w+ w+ m? w+ m2’ | 
which equations show that the locus is a bicircular quartic. 
The author then proceeds to show the existence of a third 
node, to express the locus as the inverse of a conic, and to 
exhibit it as the antipode of a conic. In the discussion on 
the paper Mr. Roberts gave some additional results bearing on 
the subject; and Mr. Spottiswoode stated that he recognised 
many of the curves (exhibited by Messrs. Cayley and Roberts) 
as having come under his notice in the course of experiments 
he had recently made with elastic strings. 
Mr. Roberts then read his paper ‘‘On the Ovals of Des 
Cartes.”” It will suffice in this notice to give the heads under 
which the author treated the subject. On the polar equation 
and its interpretation; on the description of the curves by the 
transformation of a circle; on certain systems of the curves ; 
on the normals; properties deducible from the general inter- 
pretation of the polar equation. Under the last head Mr. 
Roberts stated some interesting results: the sum of the areas 
of the Ovals—that is to say, in an obvious sense, the area of 
the Cartesian is equal to twice the area of the circle whose 
centre is at the triple focus, and which passes through the 
points of contact of the double tangent; the lengths of the 
ovals of a Cartesian are expressed by Zyzygetic relations 
between two elliptic quadrants; the difference of the lengths 
of the loops of a nodal Limagon is four times the distance 
between the vertices— Dr, Henrici exhibited a plaster cast of 
the surface, x y2— (2) (++7+2—1)*=0. 
~ 
- Royal Astronomical Society, May 13.—W. Lassell, F.R.S., 
President of the Society, in the chair. Thirty-one presents were 
announced, and the thanks of the society were voted to the respec- » 
tive donors. Prof. Selwyn read a communication from Sir John 
Herschel, referring to a chart in which the sun’s spots observed 
by Mr. Carrington during his long study of the sun have been 
referred to their proper solar longitude and latitude. Sir John 
Herschel’s object in preparing this chart was to determine what 
signs there are that any particular portions of the sun within the 
sun-spot latitude are more liable than others to be the scene of 
spot-formation. THe found very few signs of any such arrange- 
ment. Mr. Proctor remarked that Sir John Herschel’s com- 
munication seemed to dispose effectually of Prof. Kirkwood’s 
recently propounded theory in explanation of the periodicity of 
sun-spots. —Papers on the occultation of Saturn, as observed by 
Messrs, Talmage, Joynson, and Captain Noble, were then read. 
Captain Noble remarked on the singularly clear definition of 
Saturn, ‘‘right up to the moon’s limb,” as being strikingly 
opposed to the theory brought forward by Mr. Watson at the 
last meeting of the society.—A paper by Mr. Penrose, on the 
Sun-spots recently seen, led to some controversy about the real 
nature of the spots. Mr, Howlett dwelt on the exaggerated 
character of the saucer-shaped depressions usually shown in 
text-books of astronomy. Mr. De La Rue mentioned the pre- 
ponderance of evidence in favour of the spots being depressions. 
The Astronomer Royal also spoke -in favour of this view. 
Mr. Lockyer dwelt on the evidence bearing on the subject de- 
rived from his method of detailed spectroscopic examination, 
and mentioned the interesting circumstance that so far as his 
} spectroscopic observations had hitherto gone, it seemed almost as 
though the depressions were shallower now the spot period was 
nearly at its maximum than they werea year and a half ago. The 
Astronomer Royal expressed his admiration of the persistence 
and patience with which Mr. Lockyer had continued his solar 
observations, and stated his belief that in the course of the next 
two or three years we should gain much more new knowledge from 
this application of spectroscopic analysis to the sun.—At the 
President's request, Mr. Airy then referred to the expedition to 
be made to observe the eclipse of December next. He stated 
that the names of about sixty volunteers had been received. 
According to present arrangements, as many as twenty-six would 
be wanted at each station for general, spectroscopic, polariscopic, 
and photographic work. He expressed his anxiety that the ob- 
servers who were to take part in the observations on the polarisa- 
tion of the corona’s light should understand their work, and 
clearly recognise—tst, what polarisation is ; 2nd, what polarisa- 
tion in a plane means; and, 3rd, when light /s polarised in a 
plane how to recognise that plane. Mr. De La Rue spoke very 
favourably of the list of volunteers, saying that it included just 
those classes of observers which were really wanted. It was men- 
tioned that the Poet Laureate had volunteered ; and, in answer to 
a question by Prof. Grant, that good draughtsmen would accom- 
pany the expedition.—A paper by Mr. Lynn, on a star in Ursa 
Major, which has a very large proper motion, according to the 
researches of Argelander, was then read.—Mr. Proctor read a 
paper, ‘*On the Resolvability of Star-groups regarded as a 
Test of Distance,” dwelling on the evidence we have of variety 
of constitution as at least equally available to explain differences 
of resolyability as variety of distance can be. Mr. Stone said he 
thought Mr. Proctor exaggerated the theory of uniformity of 
stellar distribution ; in reality, astronomers regarded the existence 
of very various degrees of real magnitude amongstars as not only 
possible but probable, or even certain. Mr. Proctor said he was 
fully aware of that, and his views touched on quite a different 
matter. What he pointed to as wrong in accepted theories was 
that those theories failed to recognise the existence of definite 
aggregations of stars, in streams, clusters, groups, and so on, 
showing all degrees of richness of aggregation, &c.—Mr. Williams 
then described some antique telescopes of Campani’s construction, 
amazingly long, with object-glasses varying apparently from about 
half an inch to an inch in diameter. 
Zoological Society, May 12.—Prof. Newton, vice-president, 
in the chair. The secretary read some notes on the principal 
additions to the Society’s menagerie during the month of April, 
and called particular attention to a Vulturine Guinea-fowl 
(Numida vulturina), presented by Dr. John Kirk, C.M.Z.S., 
being the first living specimen of this rare species received in 
England,.-—A communication was read from Dr. R. O. Cunning- 
ham, C.M.Z.S., on some peculiarities in the anatomy of three 
kingfishers, Ceryle stellata, Dacelo gigas, and Alcedo ispida.—A 
communication was read from Mr. George Gulliver, F.Z.S., on 
the taxonomic characters afforded by the muscular sheath of the 
cesophagus in sauropsida and other vertebrates.—Mr. R. B. 
Sharpe read a paper containing a full account of the swallows 
